Since we consider domains which are \(C^{1,1}\) -regular, we can solve the mean curvature as \(\kappa = \operatorname{\mathrm {div}}_{\Gamma _{\rm f}}{\operatorname{n}} = \operatorname{\mathrm {div}}{\operatorname{N}}\)  [1]}, where \(\operatorname{N}\)  is any extension of \(\operatorname{n}\)  in \(\Omega \) . Having this in mind, we solve for \(\operatorname{N}\in H^1(\Omega )^2\)  by means of the equation
\(\varepsilon a({\operatorname{N}},{\operatorname{\varphi }})_{\Omega } + ({\operatorname{N}},{\operatorname{\varphi }})_{\Gamma _{\rm f}} = ({\operatorname{n}},{\operatorname{\varphi }})_{\Gamma _{\rm f}} \ for\ all\ \operatorname{\varphi }\in H^1(\Omega )^2.\) 
